3d vectors?

Question

if P and Q are points (-2,1,0) and (1,-5,6) respectively

are the following points on the line segment joining P and Q?
and also are they on the line through P and Q??

R ( -1 , -1 , 2 )
S ( -4 , 5 , -4 )
T ( 2 , -4 , 4 )

thanks for any help!!

Answer 1

Do u know about vector equation?
find the vector equation of PQ,then subs. in R,S,T particularly to see whether it's satisfy the vector equation of PQ or not.

Answer 2

Given the points P(-2,1,0) and Q(1,-5,6).

First find the directional vector v, of the line L thru P and Q.

v = PQ = = <1+2, -5-1, 6-0> = <3, -6, 6>

If R, S, and T are on the line L, then the directional vector between P and R, S, and T will be parallel to v. If the vector is not only parallel to v, but in the same direction and smaller than v then it is on the line segement.

PR = <-1+2, -1-1, 2-0> = <1, -2, 2>
This parallel to v so R is on the line L. Since R is also in the same direction as v and smaller it is also on the line segment PQ.

PS = <-4+2, 5-1, -4-0> = <-2, 4, 4>
This is not parallel to v so S is not on the line L.

PT = <2+2, -4-1, 4-0> = <4, -5, 4>
This is not parallel to v so T is not on the line L.